Bring the water and salt to a boil in a small saucepan. Remove from the heat, stir in the bulgur, and cover; let stand until the grains are tender, about 15 to 20 minutes. Drain the bulgur well and spread it onto a rimmed baking sheet to cool slightly—this keeps the patties from becoming gummy.

Pulse the shredded beet, walnuts, basil, and garlic in a food processor until finely chopped, about 12 pulses. Add the beans, baby food, water, mustard, salt, and pepper and pulse until well combined, about 8 pulses, while still maintaining some texture from the beans.

Transfer the beet mixture to a large bowl and stir in the panko and cooled bulgur until evenly distributed. Divide the mixture into 8 equal portions and firmly pack them into 4-inch-wide patties.

Heat half the oil in a 12-inch nonstick skillet over medium-high heat until the oil is shimmering. Gently lay 4 patties into the skillet and cook until the bottoms are crisp and well browned, about 4 to 5 minutes. Flip carefully and cook the second side until browned, reducing the heat if the burgers begin to scorch. Transfer to a plate and tent loosely with foil.

Wipe out the skillet with paper towels and repeat the frying process with the remaining oil and patties. Serve warm on toasted buns with your preferred toppings.